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   "source": [
    "# A Simple Staggered FV Code for the Navier-Stokes Equations\n",
    "### Tony Saad <br/> Assistant Professor of Chemical Engineering <br/> University of Utah"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib import cm\n",
    "plt.rcParams['animation.html'] = 'html5'\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reynolds Number: 1000.0\n",
      "dt= 0.0004\n"
     ]
    }
   ],
   "source": [
    "# define some boiler plate\n",
    "nx = 16\n",
    "ny = 16\n",
    "# dt = 1.0e-3 # timestep size\n",
    "ν = 0.01 # dynamic viscosity\n",
    "lx = 1.0\n",
    "ly = 1.0\n",
    "dx = lx/nx\n",
    "dy = ly/ny\n",
    "t = 0.0\n",
    "\n",
    "# cell centered coordinates\n",
    "xx = np.linspace(dx/2.0,lx - dx/2.0,nx, endpoint=True)\n",
    "yy = np.linspace(dy/2.0,ly - dy/2.0,ny, endpoint=True)\n",
    "xcc, ycc = np.meshgrid(xx,yy)\n",
    "\n",
    "# x-staggered coordinates\n",
    "xxs = np.linspace(0,lx,nx+1, endpoint=True)\n",
    "xu,yu = np.meshgrid(xxs, yy)\n",
    "\n",
    "# y-staggered coordinates\n",
    "yys = np.linspace(0,ly,ny+1, endpoint=True)\n",
    "xv,yv = np.meshgrid(xx, yys)\n",
    "\n",
    "Ut = 10.0\n",
    "Ub = 0.0\n",
    "Vl = 0.0\n",
    "Vr = 0.0\n",
    "print('Reynolds Number:', Ut*lx/ν)\n",
    "dt = min(0.25*dx*dx/ν, 4.0*ν/Ut/Ut)\n",
    "print('dt=', dt)\n",
    "\n",
    "# initialize velocities - we stagger everything in the negative direction. A scalar cell owns its minus face, only.\n",
    "# Then, for example, the u velocity field has a ghost cell at x0 - dx and the plus ghost cell at lx\n",
    "u = np.zeros([ny+2, nx+2]) # include ghost cells\n",
    "\n",
    "# # same thing for the y-velocity component\n",
    "v = np.zeros([ny +2, nx+2]) # include ghost cells\n",
    "\n",
    "ut = np.zeros_like(u)\n",
    "vt = np.zeros_like(u)    \n",
    "\n",
    "# initialize the pressure\n",
    "p = np.zeros([nx+2,ny+2]); # include ghost cells\n",
    "\n",
    "# a bunch of lists for animation purposes\n",
    "usol=[]\n",
    "usol.append(u)\n",
    "\n",
    "vsol=[]\n",
    "vsol.append(v)\n",
    "\n",
    "psol = []\n",
    "psol.append(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# build pressure coefficient matrix\n",
    "Ap = np.zeros([ny+2,nx+2])\n",
    "Ae = 1.0/dx/dx*np.ones([ny+2,nx+2])\n",
    "As = 1.0/dy/dy*np.ones([ny+2,nx+2])\n",
    "An = 1.0/dy/dy*np.ones([ny+2,nx+2])\n",
    "Aw = 1.0/dx/dx*np.ones([ny+2,nx+2])\n",
    "# set left wall coefs\n",
    "Aw[1:-1,1] = 0.0\n",
    "# set right wall coefs\n",
    "Ae[1:-1,-2] = 0.0\n",
    "# set top wall coefs\n",
    "An[-2,1:-1] = 0.0\n",
    "# set bottom wall coefs\n",
    "As[1,1:-1] = 0.0\n",
    "Ap = -(Aw + Ae + An + As)\n",
    "\n",
    "def pressure_poisson2(p, b, dx, dy):\n",
    "    pn = np.empty_like(p)\n",
    "    it = 0\n",
    "    err = 1e5\n",
    "    tol = 1e-8\n",
    "    maxit = 100\n",
    "    β = 1.1\n",
    "    while err > tol and it < maxit:\n",
    "        pn = p.copy()    \n",
    "        for i in range(1,nx+1):\n",
    "            for j in range(1,ny+1):\n",
    "                ap = Ap[j,i]\n",
    "                an = An[j,i]\n",
    "                aso = As[j,i]\n",
    "                ae = Ae[j,i]\n",
    "                aw = Aw[j,i]\n",
    "                rhs = b[j,i] - 1.0*(ae*p[j,i+1] + aw*p[j,i-1] + an*p[j+1,i] + aso*p[j-1,i])\n",
    "                p[j,i] = β*rhs/ap + (1-β)*p[j,i]\n",
    "        err = np.linalg.norm(p - pn)\n",
    "        it += 1\n",
    "#     print('Poisson Error:', err)        \n",
    "    return p, err"
   ]
  },
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    "# USE sparse solver\n",
    "import scipy.linalg\n",
    "import scipy.sparse\n",
    "import scipy.sparse.linalg\n",
    "# build pressure coefficient matrix\n",
    "Ap = np.zeros([ny,nx])\n",
    "Ae = 1.0/dx/dx*np.ones([ny,nx])\n",
    "As = 1.0/dy/dy*np.ones([ny,nx])\n",
    "An = 1.0/dy/dy*np.ones([ny,nx])\n",
    "Aw = 1.0/dx/dx*np.ones([ny,nx])\n",
    "# set left wall coefs\n",
    "Aw[:,0] = 0.0\n",
    "# set right wall coefs\n",
    "Ae[:,-1] = 0.0\n",
    "# set top wall coefs\n",
    "An[-1,:] = 0.0\n",
    "# set bottom wall coefs\n",
    "As[0,:] = 0.0\n",
    "Ap = -(Aw + Ae + An + As)\n",
    "\n",
    "n = nx*ny\n",
    "d0 = Ap.reshape(n)\n",
    "# print(d0)\n",
    "de = Ae.reshape(n)[:-1]\n",
    "# print(de)\n",
    "dw = Aw.reshape(n)[1:]\n",
    "# print(dw)\n",
    "ds = As.reshape(n)[nx:]\n",
    "# print(ds)\n",
    "dn = An.reshape(n)[:-nx]\n",
    "# print(dn)\n",
    "A1 = scipy.sparse.diags([d0, de, dw, dn, ds], [0, 1, -1, nx, -nx], format='csr')\n",
    "plt.matshow((A1.toarray()))"
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     "name": "stdout",
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     "text": [
      "total time 0.11582800000000049 s\n",
      "time per timestep = 0.01158280000000005 s\n",
      "mom assembly time 0.04313199999999817 s\n",
      "solver time 0.07269600000000231 s\n",
      "solver fraction = 63.0 %\n"
     ]
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   "source": [
    "# while t < tend:\n",
    "t0 = time.clock()\n",
    "momtime = 0.0\n",
    "solvertime = 0.0\n",
    "nsteps = 10\n",
    "for n in range(0,nsteps):\n",
    "    # left wall\n",
    "    u[1:-1,1] = 0.0\n",
    "    # right wall\n",
    "    u[1:-1,-1] = 0.0\n",
    "    # top wall\n",
    "    u[-1,1:] = 2.0*Ut - u[-2,1:]\n",
    "    # bottom wall\n",
    "    u[0,1:] = 2.0*Ub - u[1,1:]\n",
    "\n",
    "    # left wall\n",
    "    v[1:,0] = 2.0*Vl - v[1:,1]\n",
    "    # right wall\n",
    "    v[1:,-1] = 2.0*Vr - v[1:,-2]\n",
    "    # bottom wall\n",
    "    v[1,1:-1] = 0.0\n",
    "    # top wall\n",
    "    v[-1,1:-1] = 0.0    \n",
    "  \n",
    "\n",
    "    # do x-momentum first - u is of size (nx + 2) x (ny + 2) - only need to do the interior points\n",
    "    tic = time.clock()\n",
    "    for i in range(2,nx+1):\n",
    "        for j in range(1,ny+1):\n",
    "            ue = 0.5*(u[j,i+1] + u[j,i])\n",
    "            uw = 0.5*(u[j,i]   + u[j,i-1])    \n",
    "            \n",
    "            un = 0.5*(u[j+1,i] + u[j,i])\n",
    "            us = 0.5*(u[j,i] + u[j-1,i])            \n",
    "            \n",
    "            vn = 0.5*(v[j+1,i] + v[j+1,i-1])\n",
    "            vs = 0.5*(v[j,i] + v[j,i-1])\n",
    "            \n",
    "            # convection = - d(uu)/dx - d(vu)/dy\n",
    "            convection = - (ue*ue - uw*uw)/dx - (un*vn - us*vs)/dy\n",
    "            \n",
    "            # diffusion = d2u/dx2 + d2u/dy2\n",
    "            diffusion = ν*( (u[j,i+1] - 2.0*u[j,i] + u[j,i-1])/dx/dx + (u[j+1,i] - 2.0*u[j,i] + u[j-1,i])/dy/dy )\n",
    "            \n",
    "            ut[j,i] = u[j,i] + dt *(convection + diffusion)\n",
    "                \n",
    "    # do y-momentum - only need to do interior points\n",
    "    for i in range(1,nx+1):\n",
    "        for j in range(2,ny+1):\n",
    "            ve = 0.5*(v[j,i+1] + v[j,i])\n",
    "            vw = 0.5*(v[j,i] + v[j,i-1])    \n",
    "            \n",
    "            ue = 0.5*(u[j,i+1] + u[j-1,i+1])\n",
    "            uw = 0.5*(u[j,i] + u[j-1,i])\n",
    "            \n",
    "            vn = 0.5*(v[j+1,i] + v[j,i])\n",
    "            vs = 0.5*(v[j,i] + v[j-1,i])            \n",
    "\n",
    "            # convection = d(uv)/dx + d(vv)/dy\n",
    "            convection = - (ue*ve - uw*vw)/dx - (vn*vn - vs*vs)/dy\n",
    "            \n",
    "            # diffusion = d2u/dx2 + d2u/dy2\n",
    "            diffusion = ν*( (v[j,i+1] - 2.0*v[j,i] + v[j,i-1])/dx/dx + (v[j+1,i] - 2.0*v[j,i] + v[j-1,i])/dy/dy )\n",
    "            \n",
    "            vt[j,i] = v[j,i] + dt*(convection + diffusion)            \n",
    "    # do pressure - prhs = 1/dt * div(uhat)\n",
    "    # we will only need to fill the interior points. This size is for convenient indexing\n",
    "    divut = np.zeros([ny+2,nx+2]) \n",
    "#     for i in range(1,nx+1):\n",
    "#         for j in range(1,ny+1):\n",
    "#             divutilde[j,i] = (ut[j,i+1] - ut[j,i])/dx + (vt[j+1,i] - vt[j,i])/dy\n",
    "    divut[1:-1,1:-1] = (ut[1:-1,2:] - ut[1:-1,1:-1])/dx + (vt[2:,1:-1] - vt[1:-1,1:-1])/dy\n",
    "\n",
    "    prhs = 1.0/dt * divut\n",
    "    toc = time.clock()\n",
    "    momtime += (toc - tic)\n",
    "    \n",
    "    ###### Use the direct solver\n",
    "#     tic=time.clock()    \n",
    "#     pressure_poisson2(p,prhs,dx,dy)\n",
    "#     toc=time.clock()\n",
    "#     solvertime += toc - tic\n",
    "#     psol.append(p)\n",
    "\n",
    "    \n",
    "    \n",
    "    ###### Use the sparse linear solver\n",
    "    tic=time.clock()\n",
    "#     pt = scipy.sparse.linalg.spsolve(A1,prhs[1:-1,1:-1].ravel()) #theta=sc.linalg.solve_triangular(A,d)\n",
    "    pt,info = scipy.sparse.linalg.bicg(A1,prhs[1:-1,1:-1].ravel(),tol=1e-10) #theta=sc.linalg.solve_triangular(A,d)\n",
    "    toc=time.clock()\n",
    "    solvertime += toc - tic\n",
    "    p = np.zeros([ny+2,nx+2])\n",
    "    p[1:-1,1:-1] = pt.reshape([ny,nx])\n",
    "\n",
    "    # time advance\n",
    "    u[1:-1,2:-1] = ut[1:-1,2:-1] - dt * (p[1:-1,2:-1] - p[1:-1,1:-2])/dx\n",
    "    v[2:-1,1:-1] = vt[2:-1,1:-1] - dt * (p[2:-1,1:-1] - p[1:-2,1:-1])/dy       \n",
    "    \n",
    "        \n",
    "#     # Check mass residual\n",
    "#     divunp1 = np.zeros([ny+2,nx+2])\n",
    "#     for i in range(1,nx+1):\n",
    "#         for j in range(1,ny+1):\n",
    "#             divunp1[j,i] = (u[j,i+1] - u[j,i])/dx + (v[j+1,i] - v[j,i])/dy\n",
    "#     residual = np.linalg.norm(divunp1.ravel())\n",
    "#     if residual > 1e-6:\n",
    "#         print('Mass residual:',np.linalg.norm(divunp1.ravel()))\n",
    "    \n",
    "    # save new solutions\n",
    "#     usol.append(u)\n",
    "#     vsol.append(v)\n",
    "    \n",
    "    t += dt\n",
    "tend = time.clock()\n",
    "totaltime = tend - t0\n",
    "print('total time', totaltime, 's')\n",
    "print('time per timestep =',totaltime/nsteps, 's')\n",
    "print('mom assembly time', totaltime - solvertime, 's')\n",
    "print('solver time', solvertime, 's')\n",
    "print('solver fraction =', np.ceil(solvertime/(tend - t0)*100),'%')"
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   "source": [
    "divu = (u[1:-1,2:] - u[1:-1,1:-1])/dx + (v[2:,1:-1] - v[1:-1,1:-1])/dy\n",
    "plt.imshow(divu,origin='bottom')\n",
    "plt.colorbar()"
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  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 18,
     "metadata": {},
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   "source": [
    "fig = plt.figure(figsize=[6,6],dpi=600)\n",
    "ucc = 0.5*(u[1:-1,2:] + u[1:-1,1:-1])\n",
    "vcc = 0.5*(v[2:,1:-1] + v[1:-1,1:-1])\n",
    "speed = np.sqrt(ucc*ucc + vcc*vcc)\n",
    "plt.contourf(speed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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   "source": [
    "x = np.linspace(0,lx,nx)\n",
    "y = np.linspace(0,ly,ny)\n",
    "xx,yy = np.meshgrid(x,y)\n",
    "nn = 1\n",
    "fig = plt.figure(figsize=[6,6],dpi=600)\n",
    "plt.quiver(xx[::nn,::nn],yy[::nn,::nn],ucc[::nn,::nn],vcc[::nn,::nn])\n",
    "plt.xlim([xx[0,0],xx[0,-1]])\n",
    "plt.ylim([yy[0,0],yy[-1,0]])\n",
    "# ax.set_xlim([xx[0,0],xx[0,-1]])\n",
    "plt.streamplot(xx,yy,ucc, vcc, color=np.sqrt(ucc*ucc + vcc*vcc),density=1.5,cmap=plt.cm.autumn,linewidth=1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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